DOI 10.1007/s12220-013-9463-0
Compactness of Relatively Isospectral Sets of Surfaces
Via Conformal Surgeries
Pierre Albin· Clara L. Aldana · Frédéric Rochon
Received: 16 September 2013 / Published online: 14 November 2013 © Mathematica Josephina, Inc. 2013
Abstract We introduce a notion of relative isospectrality for surfaces with bound-ary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that allows us to reduce to the case of surfaces hyperbolic near infinity recently studied by Borthwick and Perry, or to the closed case by Osgood, Phillips, and Sarnak if there are only cusps.
Keywords Inverse spectral problem· Analytic surgery · Hyperbolic cusps · Hyperbolic funnels· Relatively isospectral
Mathematics Subject Classification 58J53· 57R65 · 58G11 · 35P20
Communicated by Steven Zelditch.
P. Albin was partially funded by NSF grant DMS-1104533. C.L. Aldana was partially funded by FCT project ptdc/mat/101007/2008. Registered at MPG, AEI-2012-059. F. Rochon was supported by grant DP120102019.
P. Albin (
B
)Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, USA e-mail:palbin@illinois.edu
C.L. Aldana
Max Planck Institut für Gravitationsphysik (AEI), Golm, Germany e-mail:clara.aldana@aei.mpg.de
F. Rochon
1 Introduction
Despite the negative answer to Mark Kac’s famous question Can one hear the shape of a drum?
there have been many interesting positive results. The line most relevant to our cur-rent investigation starts with a seminal paper by Melrose [22]. For bounded planar domains, or ‘drumheads’, the geometry is encoded in the geodesic curvature of the boundary and Melrose showed that the short-time asymptotics of the trace of the heat kernel determine this curvature to within a compact subset of the space of smooth functions.
A stronger compactness result was obtained by Osgood, Phillips, and Sarnak [26–29] by making use of the determinant of the Laplacian, in addition to the short-time asymptotics of the heat trace. Indeed, they were able to show that on a given closed surface a family of isospectral metrics is compact in the space of smooth met-rics (and similarly for planar domains). Their proof can be conveniently understood in terms of the Cheeger compactness theorem: A set of metrics with lower bounds on the injectivity radius and volume, and uniform pointwise upper bounds on the curva-ture and its covariant derivatives, is compact. The short-time asymptotics of the heat trace determine the volume and upper bounds on the Sobolev norms of the curva-ture [4,16,22,27], while for surfaces the determinant of the Laplacian gives a lower bound on the injectivity radius [27]. The control on the injectivity radius and the Sobolev norms of the curvature can then be parlayed into uniform pointwise bounds on the curvature and its covariant derivatives.
There have been many extensions of the Osgood–Phillips–Sarnak results. For in-stance, it is only recently that their result for planar domains has been extended to flat surfaces with boundary by Kim [21]. In higher dimensions there are results of Brooks–Perry–Yang [8], Chang–Yang [14], and Chang–Qing [12] studying compact-ness of isospectral metrics in a given conformal class, as well as results establishing compactness of isospectral metrics with an additional assumption on the curvature or the injectivity radius, e.g., by Anderson [3], Brooks–Glezen [4], Brooks–Perry– Petersen [7], Chen–Xu [13], and Zhou [30]. For a well-written survey of positive and negative isospectral results, we refer the reader to [17].
More recently, Borthwick and Perry [6] consider non-compact surfaces whose ends are hyperbolic funnels. For these metrics the resolvent R(s)= ( − s(1 − s))−1 admits a meromorphic continuation to the whole complex plane [18,19,24]. Two metrics whose resolvents have the same poles (with multiplicity) are called isores-onant. Borthwick and Perry prove that any set of isoresonant metrics that coincide, and are hyperbolic funnel metrics, in a fixed ‘neighborhood of infinity’ form aC∞ -compact subset in the space of metrics. This generalizes earlier work of Borthwick, Judge, and Perry [5].
One of the authors [2] has shown that metrics that are conformally equivalent to a hyperbolic surface with cusps, with conformal factors supported in a fixed com-pact set, and are mutually isoresonant, form aC∞-compact set. Note that due to the vanishing of the injectivity radius, one cannot use Sobolev inequalities to transform Sobolev bounds on the curvature to uniform bounds. We shall face the same problem below (see, e.g., Lemma2).
The results in the noncompact setting are similar in the sense that the metrics are assumed to coincide outside a compact set, but they differ in the definition of isospectrality. We propose a notion of isospectrality for noncompact manifolds that unifies these approaches. Let us say that two Riemannian manifolds (M1, g1)and
(M2, g2)coincide cocompactly if there exist compact subsets Ki⊆ Mi\ ∂Miand an isometry M1\ K1−→ M2\ K2. In this case, we say that the manifolds coincide on
U∞= Mi\ Ki. By embedding each L2(Mi, gi)into
L2(K1, g1)⊕ L2(K2, g2)⊕ L2(U∞, gi), we can consider the difference of the heat kernels
e−tg1− e−tg2.
Often this difference is trace-class even if the individual heat kernels are not. Indeed, results of Bunke [9] and Carron [10] guarantee that for complete metrics that coincide cocompactly the difference of heat kernels is trace-class.
Definition 1 Two Riemannian manifolds (M1, g1) and (M2, g2) are relatively
isospectral if they coincide on Mi \ Ki with Ki ⊂ Mi \ ∂Mi compact sets and if their relative heat trace is (defined and) identically zero,
Tre−t1− e−t2= 0 for all t > 0. (1.1)
We also say that they are isospectral relative toU∞to emphasize the neighborhood where they coincide.
If the manifolds are complete, then (1.1) can be interpreted as a condition on the ‘Krein spectral shift function’ ξ of the pair 1, 2, as one has, e.g., [10, Theo-rem 3.3]1
Tre−t1− e−t2= −
Rξ(λ)t e
−tλdλ. (1.2)
Thus with mild regularity assumptions on ξ , relatively isospectral complete metrics have ξ≡ 0, i.e., are isophasal.
As mentioned above, the metrics considered by Borthwick–Perry are relatively isospectral, as are those considered by Hassell–Zelditch, if one allows the compact sets Ki to have boundary. In the case of Borthwick–Perry this is a consequence of a Poisson formula [6, Theorem 2.1], due in this context to Borthwick, that shows that the resonance set determines the (renormalized) trace of the wave kernel, and hence the (renormalized) trace of the heat kernel. Similarly, in [20, (1.6)], it is pointed out that the scattering phase determines a renormalized trace of the heat kernel. It is straightforward to write the relative trace of the heat kernels as the difference of their renormalized traces.
Let us now introduce the type of Riemannian metrics considered here. Let M be a smooth compact surface with boundary ∂M with n marked points p1, . . . , pn∈
M\ ∂M. We suppose ∂M is the disjoint union of two types of boundaries, ∂FM
and ∂bM, each of which is a finite union of circles. Consider then the possibly non-compact surface M= M \ (∂FM∪ {p1, . . . , pn}).
Definition 2 Let M be as described above. A Funnel-cusp-boundary metric (Fc ∂-metric for short) is a Riemannian ∂-metric g on M such that:
(i) there exist a collar neighborhood
cF : ∂FM× [0, ν)x→ M,
a Riemannian metric hF on ∂FM, and a smooth function ϕF ∈ C∞(∂FM× [0, ν)x)locally constant on ∂FM× {0} such that
c∗Fg= eϕF dx2+ pr∗FhF x2 ,
where prF is the natural projection onto ∂FM;
(ii) for each marked point pi, there exist a neighborhoodVi ⊂ M, coordinates u, v onVi with u(pi)= v(pi)= 0, and a function ϕi∈ C∞(M)such that near pi,
g= eϕi du2+ dv2 r2(log r)2 , where r=√u2+ v2.
Fig. 1 (M, g) a Fc ∂ -surface
In other words, the metric is asymptotically hyperbolic near ∂FM while it is con-formal to a cusp near each marked point pi. We say (M, g) is a Fc ∂-Riemannian surface, see Fig.1.
For such a Fc ∂-metric g on M, we will consider the corresponding (positive) Laplacian g with Dirichlet boundary condition on ∂M:= ∂bM. When ∂M = ∅, Fc ∂-metrics correspond to some of the F− hc metrics considered in [1]. If ∂FM= ∅
and M has no marked points, the spectrum of gis discrete. Otherwise, each bound-ary component Y of ∂FMgives rise to a band of continuous spectrum[e−cY4 ,∞) (of
infinite multiplicity), where the constant cY is the restriction of ϕF to Y , while each marked point pi gives rise to a band of continuous spectrum[e−ϕi (pi)4 ,∞) (of
mul-tiplicity one). In particular, the continuous spectrum is bounded below by a positive constant. Notice also that 0 is in the spectrum if and only if ∂FM= ∂bM= ∅.
The main result of this paper is to establish compactness for sets of relatively isospectral Fc ∂-Riemannian surfaces. More precisely, we prove the following theo-rem.
Theorem 1 Let (Mi, gi)be a sequence of Fc ∂-Riemannian surfaces, isospectral
rel-ative toU∞. Then there is a Riemannian surface (M, g∞), a subsequence (Mik, gik),
and a sequence of diffeomorphisms
φk: M −→ Mik, with φ ◦ φ−1 |U∞= Id for any , such that the metrics φk∗gik converge to g∞inC∞.
the relative local heat invariants stay the same. From the local nature of Polyakov’s formula for the variation of the determinant, one can also hope that the relative de-terminant will remain unchanged. Using a finite speed propagation argument, we are able to show that this is indeed the case—even though for a conformal surgery at a point, the deformation does not have uniformly bounded curvature nor global Sobolev inequalities. This reduces the problem to the situations treated in [6,27], from which the compactness of the set of relatively isospectral Fc ∂-metrics follows.
The paper is organized as follows. In Sect.2we introduce conformal surgery met-rics. These are metrics depending on a parameter ε that transform the ends from cusps and funnels, at ε= 0, to disks and boundaries, at ε = 1. We also show that the non-zero spectrum of the Laplacian has a lower bound uniform in ε. The next few sections are devoted to understanding the relative trace as a function of ε,
ε→ Tre−tgε − e−thε,
when gε and hε are two Fc ∂-metrics that coincide along the ends and are both un-dergoing conformal surgery. Section3is devoted to establishing some L2-estimates on the heat kernel of a Fc ∂-metric, which we use in Sect.4to show that
e−tgε− e−thεTr→ 0 exponentially, as t → ∞,
and in Sect.5to show that
Tre−tgε− e−thε− Tre−tg0− e−th0 →0 exponentially, as t→ 0,
and ultimately that the relative trace is continuous in ε all the way down to ε= 0. These results are strong enough, as we point out in Sect. 6, to conclude that the relative determinant of gεand hεis also continuous down to ε= 0.
Now assume that there is some value of the parameter ε where the metrics gεand
hεare relatively isospectral. At this value the relative heat invariants and the relative determinant all vanish and, since these invariants have local transformations under conformal changes of the metrics, it follows that they must vanish for all values of ε. Thus by a conformal surgery we can transform any two relatively isospectral Fc ∂-metrics into two ∂-metrics, on either a closed manifold or one with funnel ends, with vanishing relative heat invariants and relative determinant. In Sect.7we point out that this is enough to run the proof of compactness of [6,27], and this in turn is enough to conclude compactness of the class of relatively isospectral Fc ∂-metrics.
2 Conformal Surgeries
Let (M, g) be a Fc ∂-Riemannian surface. Fix a marked point p∈ M \∂M and choose coordinates u and v in a neighborhoodV of p such that u(p) = v(p) = 0 and such that in these coordinates the metric takes the form
Fig. 2 Conformal surgery at a
point p
where we are using the polar coordinates u= r cos θ, v = r sin θ, and ρ =log r−1. With-out loss of generality, we can assume that the neighborhoodV is given by the open disk of radius r= 45. Let σ ∈ Cc∞(V) be a function taking values between 0 and 1
with σ≡ 1 for r <14 and σ≡ 0 for r >12. Consider then the function ψ(ε, r) given by ψ (ε, r)= σ (r) r2(log r)2 ε2+ (ε2+ r2)(log√r2+ ε2)2 +1− σ (r). (2.1) Extending ψε(r)= ψ(ε, r) by 1 outside the neighborhood V allows us to consider
the family of metrics
gε= ψεg.
For ε > 0, this metric is smooth at the marked point p. However, as ε approaches 0, the point p is pushed to infinity so that in the limit we recover the metric g. We say the metric g undergoes a conformal surgery at the marked point p, see Fig.2
Similarly, suppose now that ∂M is nonempty and fix a boundary component ∂iM. Instead of a conformal surgery at pi, we can consider one at ∂iM. Thus, we now take
V ∼= ∂iM× [0, 1)r to be a collar neighborhood of ∂iMin M and let θ be the angular variable on ∂iM ∼= S1. Near the boundary, the metric is of the form
g= efdr2+ dθ2, f ∈ C∞(V).
InV, consider a function ψ(ε, r) such that ε2+ r2ψ (ε, r)∈ C∞(V), ψ (ε, r)= 1 if ε > 1/2 or r > 1/2, ε2+ r2ψ (ε, r)= ew if ε2+ r2≈ 0, (2.2)
where w∈ C∞(V) is such that w + f is constant on ∂iM. For instance, we can take
w= −f . Extending ψε(r)= ψ(ε, r) by 1 outside the neighborhood V allows us to
consider the family of metrics
gε= ψεg.
In this case, the boundary component ∂iMis pushed to infinity as ε approaches zero, so that in the limit the metric becomes asymptotically hyperbolic in that end,
g0= ew+f
dr2+ dθ2
r2 , rsmall.
Fig. 3 Conformal surgery at a boundary component ∂iM
To study the determinant, it will be important for us to know that the spectra stay away from zero under conformal surgeries at a point or at a boundary component. Theorem 2 Let (M, g) be a Fc ∂-Riemannian surface. Let the family gεbe a confor-mal surgery at point p or at boundary component ∂iM. Then the smallest non-zero
eigenvalue of gε is bounded below by a constant c > 0 independent of ε.
Proof Suppose first that gε is a conformal surgery at a point p. Let λε>0 be the smallest positive eigenvalue of gε and let u∈ C∞(M)∩ L
2(M, g
ε)be the corre-sponding eigenfunction, so that gεu= λεu. When ε > 0, notice that u∈ C∞(M),
so its restriction to M\ {p} will be in L2(M, g0), in fact, in the domain of g0. Now,
we can find a constant C > 0 such that
dvolgε≤ C dvolg0, ∀ε ≥ 0.
On the other hand, since our change of metric is conformal, du2L2(M,g
ε) =
du2 L2(M,g
0). Thus, if zero is not in the spectrum, that is, if (M, g) has at least one
funnel end or if it has a non-empty boundary, we have that
λε= du2 L2(M,g ε) u2 L2(M,g ε) ≥ 1 C du2 L2(M,g 0) u2 L2(M,g 0) ≥λ0 C,
where λ0= inf Spec(0). It therefore suffices to take c=λ0
C in this case. If instead
(M, g)has an empty boundary and no funnel end, we know that zero is an eigenvalue.
Thus, in this case, we have instead
where Π runs over 2-dimensional subspaces ofCc∞(M). Since Cc∞(M)is densely contained in the domain of g0, we have again
λε= inf Π maxv∈Π dv2 L2(M,g ε) v2 L2(M,g ε) ≥ 1 CinfΠ maxv∈Π dv2 L2(M,g 0) v2 L2(M,g 0) ≥λ0 C,
where λ0= inf(Spec(0)\ {0}), so that we can still take c =λ0
C to obtain the result. If instead gε is a conformal surgery at a boundary component ∂iM, notice that if
u∈ C∞(M)is an eigenfunction of gε for ε > 0, then, since u is zero on ∂iM, it is in L2(M, g
0), in fact in the domain of g0. On the other hand, we can clearly find a
constant C > 0 such that
dvolgε≤ C dvolg0 ∀ε ∈ [0, 1].
With these two facts, we can proceed as before to obtain the result.
3 L2-Estimates for the Heat Kernel
In [11], some L2estimates are obtained for the heat kernel using a finite speed prop-agation argument. More precisely, they obtained a bound on the norm of the heat kernel acting on L2-functions. As observed by Donnelly in [15], using the Sobolev embedding, this also gives an estimate for the Hilbert–Schmidt norm of the heat ker-nel. Since we will use this observation several times, we will, for the convenience of the reader, go through this argument in detail.
Lemma 1 Let (M, g) be a Fc ∂-Riemannian surface. LetU and G be two open sets
in M and let d= dg(U, G) ≥ 0 be the distance between them. Suppose the closure of
Gis compact in M\ ∂M. If d > 0 there exists a constant CGdepending on G such that
U
e−tg(x, y)e−tgx, ydx≤ CGe−d28t ∀y, y∈ G, ∀t > 0.
If instead d= 0 then, given ν > 0, there exists a constant CG,ν>0 such that
U
e−tg(x, y)e−tgx, ydx≤ CG,ν ∀y, y∈ G, ∀t ≥ ν.
Proof Suppose first that d > 0. Let Gbe an open set relatively compact in M such that G⊂ Gwith ∂ Gsmooth and d= dg(U, G) >√d
2. Let χ∈ C
∞
c (G)be a nonneg-ative cut-off function with χ≡ 1 in a neighborhood of G. On G× G, consider the distribution
Wy, y:=
Ue
The reason for inserting the cut-off function χ in the definition of W is to be able to integrate by parts later on.
To prove the lemma, we will first show that the distributions k G,y
G,yW (y, y
)
are in L2, where k, ∈ N and Gis the Laplacian of the metric g on Gwith Dirichlet boundary conditions. Let{ui}, i ∈ N, be an orthonormal basis of L2(G, g)given by eigenfunctions of G and let 0 < λ1≤ λ2≤ · · · be the corresponding eigenvalues. Then{ui(y)uj(y)}, i, j ∈ N is an orthonormal basis of L2(G× G, g× g). On G, we know from the Weyl law that there exists a constant c > 0 such that λj∼ cj as j tends to infinity, so that
K:= i,j 1 λ2iλ2j <∞.
On the other hand, by the L2-estimate of [11, Corollary 1.2] applied to the heat kernel, we know that, for any k, ∈ N, there exists a constant Ck, >0 such that
G×Gk G,y G,yW y, yui(y)uj ydydy ≤ Ck, e− d2 4t < Ck, e−d28t, (3.2)
for all i, j∈ N, and t > 0. Now, given v ∈ L2(G× G, g× g), it can be written as vy, y= i,j μijui(y)uj y, withv2L2= i,j |μij|2.
Using (3.2), we can pair it with k G,y G,yW (y, y ), namely, G×G k G,y G,yW y, yvy, ydydy = G×G k+1 G,y +1 G,yW y, y−1 G,y −1 G,yv y, ydydy (3.3) = i,j μij λiλj G×G k+1 G,y +1 G,yW y, yui(y)uj ydydy (3.4) ≤ 1 λ2iλ2j 1 2 i,j |μij|2 1 2 Ck+1, +1e− d2 8t (3.5) = KvL2Ck+1, +1e− d2 8t, ∀t > 0. (3.6)
We can thus apply the Sobolev embedding theorem to bound the C0-norm of W on
G× G, giving the desired estimate. Notice also that we have shown W is smooth on
G× G.
If instead d= 0, we can follow the same argument, except that instead of (3.2), there is a constant Ck, ,ν depending on ν > 0 such that
G×Gk G,y G,yW y, yui(y)uj ydydy ≤ Ck, ,ν, ∀i, j ∈ N, ∀t ≥ ν.
For some of the applications of this estimate, we will allow the open set G to move towards infinity and it will be essential to understand how the constant CG grows under such a change. From the proof Lemma1, this constant depends in a subtle way on the eigenvalues of the Laplacian G. To control the growth of CG, we will instead derive the estimate, using the fact (established in the proof of Lemma1) that W is smooth. This requires some preparation.
Consider the hyperbolic metric
gH2=
dx2+ dy2 y2
on the upper half-planeH2= {z = x + iy ∈ C; y > 0}. The hyperbolic cusp metric (also called the horn metric) is obtained from this metric by taking the quotient of H2by the isometric action ofZ generated by z → z + 1. By making the change of coordinates u= log y, we can write the hyperbolic cusp metric as
ghc= du2+ e−2udx2 on H= R × (R/Z).
With respect to the change of variable x= θ and ρ = e−u, we see this simply cor-responds to the metric dρ2
ρ2 + ρ
2dθ2considered in the conformal surgery at a point. However, for the study of the constant CG, it will be more convenient to work with the coordinates (u, x). For a > 0, consider the open set
Ua=
(u, x)∈ R × (R/Z) : u < a.
Lemma 2 Let W∈ C∞(H× H ) be such that there exist constants Ca,k,ldepending on a, k, and l such that
U a×Ua Wy, ykg hcu(y) l ghcv ydydy ≤ Ca,k,luL2(U a,ghc)vL2(Ua,ghc)
for all u, v∈ L2(Ua, ghc)with supports compactly included inUa. Then there exists
a constant C independent of a, k, and l such that
Proof Notice first that for y0, y0 inUa, we have that Wy0, y0 = Ua×Ua Wy, yδy0(y)δy0 ydydy, (3.7)
where δy0 is the Dirac delta function centered at y0. Let ψ ∈ C(H
2)be a cut-off function taking values between 0 and 1 such that
ψ (u, x)=
1, 0≤ x ≤14 and u≤ 0, 0, |x| ≥12or u≥ 1.
Then define the translated function ψa,b(u, x)= ψ(u − a, x − b). Clearly, there is a
constant K independent of a and b such that sup|H2ψa,b| ≤ K e2a+ 1, sup|∇ψa,b|g H2≤ K ea+ 1, sup|ψ| = 1. (3.8) Let q: H2→ H2/Z be the quotient map. Given y0∈ Ua, choosey0∈ H2such that
q(y0)= y0and let uy0 be the unique solution in the Sobolev space L
2
1(H2, gH2)to δy0=
3
H2uy0.
By symmetry, there is a constant K1independent ofy0∈ H2such that uy0L21(H2,g
H2)≤ K1. (3.9)
Take b:= x(y0)to be the x coordinate ofy0so that ψa,b≡ 1 in a neighborhood of y0. Using this cut-off function, this means the equality
δy0=
3 j=0
jH2αj (3.10)
descends to an equality onUa+1, where
α0= −(H2ψa,b)2H2uy0+ 2∇ψa,b· ∇(H2uy0) α1= −(H2ψa,b)H2uy0− ψa,b
2
H2uy0+ 2∇ψ · ∇(H2uy0) α2= −(H2ψa,b)uy0+ 2∇ψa,b· ∇uy0
α3= ψa,buy0.
(3.11)
By (3.9),α3L2(H2,g
H2)≤ K1. On the other hand, since∇ψa,b is supported away
fromy0, notice by elliptic regularity and using (3.8) and (3.9) that there is a positive constant K2independent of a andy0such that
αjL2(H2,g
H2)≤ K2
Thus, plugging (3.10) into (3.7) and using the hypothesis of the lemma, we obtain sup Ua×Ua |W| ≤ Ce4a+ 1 3 k,l=0 Ca+1,k,l
for a constant C depending only on K, K1, and K2.
4 Long Time Behavior of the Relative Trace
As mentioned in the Introduction, the crux of the proof of compactness of relatively isospectral metrics is showing that the relative determinant of the Laplacian and the relative heat invariants are unchanged by conformal surgery. The Laplacian is confor-mally covariant in dimension two, so this will follow from knowing that the variation of these invariants is continuous, so ultimately from showing that the relative trace of the heat kernel is uniformly continuous along a conformal surgery. This section provides one of the ingredients for such a result, namely, a good uniform control of the relative trace for large time.
Fix a family of functions ψε as in (2.1) or (2.2) with support in an open setV containing the point or the boundary component at which the conformal surgery is performed. Let g and h be two Fc ∂-metrics on M such that h= g on U ∪ V, where
U = M \ K and K is a compact set. For these metrics, we can consider the
corre-sponding conformal surgeries gε = ψεg and hε= ψεh. Clearly, for all ε > 0, we have that hε= gε onU ∪ V, while for ε = 0, we have that h0= g0on (U ∪ V) \ {p} for a conformal surgery at a point p, and h0= g0on (U ∪ V) \ ∂iMfor a conformal surgery at a boundary component ∂iM. By the results of Bunke [9] and Carron [10], we know the difference of heat kernels
e−tgε− e−thε (4.1)
is trace-class for all ε≥ 0 and all t > 0. We will in fact need a more precise statement about the trace norm of this difference of heat kernels.
Proposition 1 Given T > ν > 0, there exists a positive constant C such that
e−tgε− e−thεTr≤ C ∀ε ∈ [0, 1], ∀ t ∈ [ν, T ].
Proof For metrics that coincide outside a compact set, Bunke [9] established a bound on the trace norm of the relative heat kernel. By partially following his proof, but also adding a new twist, we will show that this bound can be achieved uniformly in ε.
Let S be the point or the boundary component where the conformal surgery is taking place. Let us choose two open sets in M,W1, andW2, such that
(i) M= W1∪ W2;
(iii) gε|W1= hε|W1 for all ε∈ [0, 1];
(iv) gε|W2= g0|W2and hε|W2= h0|W2 for all ε∈ [0, 1].
We will also chooseW1andW2so that they are contained in bigger open sets W1 and W2satisfying the same properties and such that
gε= hε= g = h on W1\ W1 and W2\ W2, ∀ε ∈ [0, 1].
In [9], the strategy is to estimate the trace-norm of the difference of the heat kernels by considering the restriction of the difference toW1× W1,W2× W2,W1× W2, andW2× W1, and then applying finite propagation speed estimates.
Let us consider first the regionW2× W2. Let φ∈ Cc∞(M) be a function with
φ≡ 1 near W2 and supp φ ⊂ W2. Let χ∈ Cc∞(M)be another function such that
χ≡ 1 on supp φ and supp χ ⊂ W2. Finally, let γ∈ C∞(M)be a function with γ ≡ 1 on supp(1− φ) and γ ≡ 0 on W2. Consider then the approximate heat kernel
Hε(t, x, y)= γ (x)e−tg0(x, y)
1− φ(y)+ χ(x)e−tgε(x, y)φ (y).
Since limt→0Hε(t, x, y)= δx,y, we obtain via Duhamel’s principle that
e−tg0(x, y)− Hε(t, x, y) = − t 0 M\S e−sg0(x, z) ∂ ∂t + g0 Hε(t− s, z, y)dzds.
Thus, for x and y inW2, we have that Hε(t, x, y)= e−tgε, so that
e−tg0(x, y)− e−tgε(x, y)= − t 0 G e−sg0(x, z)Eε(t− s, z, y)dzds,
where G⊂ W2\ W2is the support of dχ and
Eε(t, z, y)= (gχ )(z)e−tgε(z, y)− 2
∇zχ (z),∇ze−tgε(z, y)
g.
If PGand PW2 are the projection operators obtained by multiplying by the charac-teristic functions of G andW2, this can be rewritten as
PW2e−tg0− e−tgεP W2= − t 0 PW2e−sg0PGPGEε(t− s)PW 2 ds. (4.2) If d= distg(G,W2)is the distance between G andW2with respect to the metric g, then using the L2estimates of [11], we know by Lemma1(cf. [9, p. 69]) that there exists a positive constant C depending on G such that
PW2e−sg0PGHS≤ Ce− d2 8s, PGEε(t− s)PW 2HS≤ Ce − d2 8(t−s),
where · HSis the Hilbert–Schmidt norm. Since we haveABTr≤ AHSBHS for two Hilbert–Schmidt operators A and B, we see from (4.2) that
PW2e−tg0 − e−tgεP
for a positive constant C independent of t∈ (0, T ] and ε ∈ [0, 1]. We obtain similarly that
PW2e−th0 − e−thεP
W2 Tr≤ C. Combining these two inequalities, this means that for t∈ [ν, T ],
PW2e−tgε− e−thεP
W2 Tr≤ 2C + maxt∈[ν,T ] PW2
e−tg0− e−th0PW
2 Tr, giving the desired uniform bound in ε in that region.
For the regionW1× W1, Bunke writes the difference of heat kernels as a sum of products of Hilbert–Schmidt operators whose norm is bounded by Ce−d28t using a
finite speed propagation argument; see [9, Theorem 3.4]. Here, the positive constants
Cand d depend on a choice (independent of ε) of compact region G in W1\ W1and on the metric h= hε= gε= g in that region. In particular, the constants C and d can be chosen to be the same for all ε∈ [0, 1].
For the regionsW1× W2, we cannot proceed as in [9], since we do not have a uniform bound on the curvature in ε for a conformal surgery at a point. Instead, we will use a finite speed propagation argument. Since we already control the trace norm onW1× W1andW2× W2, it suffices to control the trace norm in a smaller open set
W1× W2⊂ W1× W2, where the open sets W1and W2can be chosen to be disjoint and such that
M\ W2⊂ W1⊂ W1 and M\ W1⊂ W2⊂ W2,
with gε= hε= g0= h0onW1\ W1andW2\ W2. Let φ1, φ2∈ C∞(M)be nonnegative functions such that (i) φi≡ 1 in an open neighborhood of Wi for i= 1, 2; (ii) supp φ1∩ supp φ2= ∅.
Since the kernel
Eε(t, x, y)= φ1(x)
e−tgε(x, y)− e−thε(x, y)φ
2(y)
is supported away from the diagonal, its limit as t→ 0 is zero. Thus, by Duhamel’s principle, we have that
Eε(t, x, y)= t 0 M\S e−sgε(x, z) ∂ ∂t + gε Eε(t− s, z, y)dzds. Using the fact that gε= hεonW1and supp φ1⊂ W1, we see that this can be rewritten as Eε(t, x, y)= t 0 G e−sgε(x, z)G ε(t− s, x, y)dzds, where G⊂ W1\ W1is the support of dφ1and
If PW
1, PW2, and PG are the projection operators obtained by multiplying by the
characteristic functions of W1, W2, and G, then this can be reformulated as
PW 1 e−tgε− e−thεP W2= t 0 PW 1e −sgεP G PGGε(t− s)PW2 ds.
Then, for d:= mini=1,2distg(G, Wi) >0, we see from Lemma1that there is a posi-tive constant C depending on G such that
PW1e
−sgεPGHS≤ Ce−d28s, PGGε(t− s)P
W2HS≤ Ce
− d2
8(t−s).
We can thus conclude as before that there is a constant C1>0 such that PW
1
e−tgε− e−thεP
W2 Tr≤ C1, ∀t ∈ [0, T ], ∀ε ∈ [0, 1]. (4.4) Finally, for the regionW2× W1, we also only need to control the trace norm on W2× W1. Since PW 2(e −tgε − e−thε)P W1 is the adjoint 2 of P W1(e −tgε − e−thε)P
W2, the desired estimate follows from (4.4) in this case.
Using Theorem2and Proposition1, we obtain the following estimate for the be-havior of the relative trace as t tends to infinity.
Corollary 1 Let μ > 0 be a uniform lower bound for the positive spectrum of gε
and hε for all ε∈ [0, 1]. Then there exist T > 0 and K > 0 independent of ε such
that
e−tgε − e−thεTr≤ Ke−μ2t, ∀t ≥ T .
Proof Recall that any Fc ∂ metric has a punctured neighborhood of 0 disjoint from
its spectrum. Assume first that 0 is not in the spectrum. Since there is a constant
C≥ 1 such that gε
C ≤ hε≤ Cgε for all ε, we know by the spectral theorem that for
t0=log(2C 2) μ , we have e−t0gε ≤1 2, e −t0hε ≤1 2,
where · is the operator norm defined with respect to the norm of L2(M, gε). For
t≥ 2t0, notice that e−tgε− e−thεTr = e−2tgεe−t2gε− e−2thε+e−2tgε− e−2thεe−t2hε Tr ≤e−t 2gε + e−2thεe−t2gε− e−2thεTr ≤ e−2tgε− e−2thεTr. (4.5)
Applying this inequality finitely many times, we see that for all t≥ 2t0, e−tgε− e−thεTr≤ max
τ∈[t0,2t0]
e−τgε− e−τhεTr.
By Proposition1, this means the relative trace is uniformly bounded for t≥ 2t0 and ε∈ [0, 1]. Now, using the fact that e−tgε ≤ e−tμande−thε ≤ ce−tμfor
some constant c > 0 depending only on g and h and proceeding as in (4.5), we thus have for t≥ 2t0,
e−tgε− e−thεTr≤e−t2gε + e−2thεe−t2gε− e−2thεTr
≤ (1 + c)e−μ2t max
τ∈[t0,2t0]
e−τgε− e−τhεTr. (4.6)
By Proposition1, maxτ∈[t0,2t0]e−τgε − e−τhεTris bounded above by a positive
constant independent of ε. This gives the desired result. If zero is in the spectrum, we can obtain the same result by first projecting off the constants.
5 Finite Time Behavior of the Relative Trace
To introduce and study the relative determinant, we need to obtain some good control on the relative trace as ε tends to 0. We will adapt the methods used in [15] to show that the relative heat trace is continuous in ε for small t .
Theorem 3 For T > 0, the functional
ε→ Tre−tgε− e−thε
is continuous at ε= 0 uniformly with respect to t ∈ (0, T ].
Intuitively, we expect Theorem3to hold from the fact that the singular behavior of gεand hεshould cancel out. What allows us to turn these local considerations into a statement about the heat kernels is a finite propagation speed argument. We will proceed in three steps. We will use the notation S to denote either{p} or ∂iM de-pending on whether we are considering a conformal surgery at a point or at boundary component.
Lemma 3 There exist constants d > 0 and C > 0 such that Tr
e−tgε − e−thε− Tre−tg0− e−th0< Ct e−d28t
for all t > 0 and ε∈ [0, 1].
Proof Note that we expect such a rapid decay as t tends to zero from the fact that the
LetW1, W1,W2, and W2be open sets as in the proof of Proposition1. To estimate the difference of relative traces onW1, let φ∈ C∞(M)be a function with φ≡ 1 near
W1 and supp φ⊂ W1. Let χ∈ C∞(M)be another function with χ≡ 1 on supp φ and supp χ⊂ W1. Let also γ ∈ Cc∞(M)be a function with γ ≡ 1 on supp(1 − φ). Consider then the approximate heat kernel
E(t, x, y)= γ (x)e−tgε(x, y)1− φ(y)+ χ(x)e−thε(x, y)φ (y).
By Duhamel’s principle, we have that
e−tgε(x, y)− E(t, x, y) = − t 0 M\S e−sgε(x, z) ∂ ∂t + gε E(t− s, z, y)dzds. (5.1)
If we assume now that x and y are equal and lie in W1, then Eε(t, x, x)= e−thε(x, x), so proceeding as in [15, Proposition 5.1], we have that
W1\S |e−tgε(x, x)− e−thε(x, x)|dx ≤ C t 0 G W1\S |e−sgε(x, z)|2dx 1 2 W1\S |∇ze−(t−s)hε(z, x)|2dx 1 2 + W1\S |e−(t−s)hε(z, x)|2dx 1 2 dzds, (5.2)
where C > 0 is a constant depending on the norm of dχ and hχand G⊂ W1\ W1 is a compact set containing the support of dχ . By Lemma1, we have
W1\S |e−sgε(x, z)|2dx≤ C1e−d28s, W1\S |e−(t−s)hε(z, x)|2dx≤ C1e−8(td2−s), W1\S |∇ze−(t−s)hε(z, x)|2dx≤ C1e−8(td2−s), (5.3) where C1is a constant depending on G, so is independent of ε, while d > 0 is chosen to be smaller than the distance betweenW1and supp dχ with respect to the metric h (recall that h= hε= gε= g on W1\ W1). Combining (5.2) and (5.3), we obtain
W1\S
|e−tgε(x, x)− e−thε(x, x)|dx < 2 Vol(G)CC1t e−d28t ∀ε ∈ [0, 1], ∀ t > 0.
Using the fact that gε= g0 and hε= h0 on W2 and that gε= hε = g = h on
W2\ W2, we can proceed in a similar way to obtain that W2 |e−tgε(x, x)− e−tg0(x, x)|dx < Cte−d28t, W2 |e−thε(x, x)− e−th0(x, x)|dx < Cte−d28t, (5.5)
for all ε∈ [0, 1] and t > 0, for potentially different constants C > 0 and d > 0. Fi-nally, combining (5.4) and (5.5) gives the result.
An easy consequence of Lemma3is the following.
Corollary 2 The relative heat invariants in the asymptotic expansion
Tre−tgε− e−thε∼ t−1
k≥0
ak(gε, hε)tk (5.6)
are preserved under conformal surgery, i.e., ak(gε, hε)= ak(g0, h0)for all k and
all ε.
Lemma 4 Given δ > 0 and T > 0, there exist an open set A⊂ V containing S and
ε0>0 such that for all t < T and all ε∈ [0, ε0),
A\S
|e−tgε(x, x)− e−thε(x, x)|dx < δ.
Proof This is a finite propagation speed argument as in [15, Proposition 5.1], namely, we consider the approximate heat kernel
E(t, x, y)= γ (x)e−tgε(x, y)1− φ(y)+ χ(x)e−thε(x, y)φ (y), (5.7)
where φ, χ , and γ are smooth functions on M with φ≡ 1 near p and φ ≡ 0 when r > 1
2 onV (and more generally outside V), χ ≡ 1 when r ≤
1
2 and χ≡ 0 when r > 3 4, and γ ≡ 1 on supp(1 − φ) and supp γ is disjoint from p. Let G be a compact set containing the support of dχ and choose A⊂ V such that φ ≡ 1 on A. By choosing
Aand ε0>0 sufficiently small, we can ensure that dgε(A, G) > dwhen ε∈ [0, ε0],
where C is a constant depending on G. By Lemma1and using the fact that gε= g and hε= h near G, we have
A\S |e−tgε(x, z)|2dx≤ C1e−d28t, A\S |e−thε(z, x)|2dx≤ C1e−d28t, A\S |∇ze−thε(z, x)|2dx≤ C1e−d28t, (5.9)
where the positive constant C1depends only on G and is thus independent of ε. By taking A and ε0sufficiently small, we can make d as large as we want. From (5.8), given δ > 0 and T > 0, we can thus choose A and ε0so that
A\S
|e−tgε(x, x)− e−thε(x, x)|dx < δ,
for all ε∈ [0, ε0] and t ∈ (0, T ].
We need also to control the trace on the complement of A.
Lemma 5 Let N⊂ M be an open set with N ∩ S = ∅. Then given δ > 0 and T >
ν >0, there exists ε0>0 such that for all ε∈ [0, ε0] and t ∈ [ν, T ], N |e−tgε(x, x)−e−tg0(x, x)|dx < δ, N |e−thε(x, x)−e−th0(x, x)|dx < δ.
Proof We will prove the lemma for the metric gε, the proof being the same for the metric hε. Without loss of generality, by taking N bigger if needed, we can assume
M\ N is contained in the neighborhood V. This time, we consider φ ∈ C∞(M)with
φ≡ 1 on N and φ ≡ 0 near S, χ ∈ C∞(M)with the same properties and such that
χ≡ 1 on the support of φ, and we choose a function γ ∈ Cc∞(V) with γ ≡ 1 on
supp(1− φ). With these functions, we define the approximate heat kernel
E(t, x, y)= γ (x)e−tg0(x, y)1− φ(y)+ χ(x)e−tgε(x, y)φ (y). (5.10)
Using Duhamel’s principle, we have
e−tg0(x, y)− E(t, x, y) = − t 0 M\S e−sg0(x, z) ∂ ∂t + g0 E(t− s, z, y)dzds. (5.11) Suppose now that x and y are equal and lie in N . Then E(t, x, x)= e−tgε(x, x).
Writing gε= eϕεg0, we also have
Thus, from (5.11), when x and y are equal and lie in N , we have e−tg0(x, x)− e−tgε(x, x) = 2 t 0 G e−sg0(x, z)∇zχ ,∇ze−(t−s)gε(z, x)dzds − t 0 G e−sg0(x, z)g 0χ (z) e−(t−s)gε(z, x)dzds + t 0 G e−sg0(x, z)1− eϕεχ (z) gεe−(t−s)gε(z, x)dzds, (5.13)
where G= supp dχ ⊂ M \ S is a compact set and G⊂ V \ S is a compact set con-taining supp χ∩ supp(1 − eϕε).
When we integrate with respect to x on N , the first two terms on the right-hand side of (5.13) can be bounded as before by
C t 0 G N |e−sg0(x, z)|2dx 1 2 N |e−(t−s)gε(z, x)|2dx 1 2 + N |∇ze−(t−s)gε(z, x)|2dx 1 2 dzds,
where C is a constant depending on the norm of dχ and g0χon G with respect to the norm of gε. By Lemma1, we can bound this expression by
CVol(G, g0)Vol(G, gε)CG,g0CG,gεe− d20 +d2ε
16t , (5.14)
where dε is the distance between G and N with respect to the metric gε, C is a constant depending on C and ν, and CG,gε is the optimal constant for the estimate of
Lemma1for gεon G.
In the case of a surgery at a boundary component, using the fact that g0is quasi-isometric to a Fc ∂-metric which is hyperbolic near infinity, we see using Lemma2
with the constant a fixed that we can take CG,g0 to be independent of G. On the other
hand, for fixed G, we can take the constant CG,gε to be as close as we want to CG,g0
by taking ε sufficiently small. Now, at the cost of changing χ and taking ε sufficiently small, we can make d0 and dε as large as we want. Choosing χ to be independent of θ near p, this can be achieved while keeping the norm of dχ bounded by a fixed constant K with respect to g0, so that its norm with respect to gεwill be bounded by
K+ 1 if ε is small enough. During such a procedure, the volume of G with respect to g0satisfies an estimate of the form Vol(G, g0)≤ C0ed0 for some fixed constant C0, and again, for fixed G, by taking ε small enough, we can make Vol(G, gε)as close as we want to Vol(G, g0). This means that by choosing χ and ε suitably, we can make (5.14) as small as we want.
Sobolev constant of g0on G satisfies an estimate of the form
CG,g0≤ K
e4d0+ 1
for some constant K. Thus, by choosing G sufficiently far away from N , we can make the term CG,g0e−
d20
8t as small as we want. Since a cusp end has finite area, this
can be done in such a way that the volume of G with respect to gεis bounded above by a constant independent of d0and ε.
Since on the other hand we can, for fixed G, make CG,gεarbitrarily close to CG,g0
by taking ε sufficiently small, we see we can again make (5.14) as small as we want by taking G sufficiently far away from N and ε sufficiently small.
For the third term in (5.13), we note that its integral with respect to x on N can be bounded by C t 0 G N |e−sg0(x, z)|2dx 1 2 N |gεe−(t−s)gε(z, x)| 2dx 1 2 1−eϕε(z)dzds. (5.15) Since we are assuming t≥ ν > 0, each integral in x can be uniformly bounded (for G fixed) using Lemma1with d= 0. Thanks to the term (1−eϕε), the overall expression
can be made arbitrarily small by taking ε > 0 sufficiently small. These three lemmas can then be combined to give the proof of Theorem3.
Proof of Theorem3 Given T > 0 and δ > 0, we need to find ε0>0 such that when
ε∈ [0, ε0],
M\S
e−tgε(x, x)− e−thε(x, x)−e−tg0(x, x)− e−th0(x, x)dx < δ
∀ t ∈ (0, T ].
By Lemma3, we can find ν > 0 such that this integral is smaller than δ3 for t≤ ν. By Lemma4, we can find ε0>0 and an open set A⊂ V containing S such that the integral restricted to A\ S is smaller than δ3 when ε∈ [0, ε0] and t ∈ (0, T ]. On the other hand, choosing an open set N⊂ (M \ S) containing the complement of A, we know by Lemma5 that by taking ε0smaller if needed, we can ensure the integral restricted to N is also bounded by 3δ for ε∈ [0, ε0] and t ∈ [ν, T ], from which the
result follows.
6 The Relative Determinant
By Corollaries1and2, the relative zeta function given by
is well defined for Re s > 1. In fact, using the short-time asymptotic expansion (5.6), the relative zeta function can be extended meromorphically to s∈ C with at worst simple poles in s, but with s= 0 a regular point. Thus, a relative determinant can be defined by det(gε, hε)= exp −ζ(gε, hε,0) . (6.1)
Lemma 6 For ε > 0, we have
d
dεdet(gε, hε)= 0.
Proof Using the regularized trace as in [1], we can write the relative trace as a dif-ference of two regularized traces. Similarly, we can write the relative determinant as a quotient of two regularized determinants. Therefore,
log det(gε, hε)= log
R
det(gε)− log
R
det(hε).
Applying the Polyakov formula of [1] to this ratio of regularized determinants, we see that the contribution coming from one regularized determinant is canceled by the
other, from which the result follows.
Remark 1 In [1] the surfaces considered have no boundary, but since the discussion about the regularized trace and the Polyakov formula is local near the cusps and the funnels, the extension to the boundary case is automatic. As in [28], one simply needs to add an extra term in the Polyakov formula of [1] expressed in terms of the geodesic curvature of the boundary.
This immediately gives the following.
Theorem 4 For the families of metrics gεand hε, the relative determinant det(gε,
hε)is independent of ε.
Proof By Corollary1, Lemma3, and Theorem3, the relative determinant is a con-tinuous function of ε. On the other hand, by Lemma6, it is constant for ε > 0. By
continuity, it is therefore constant for ε≥ 0.
We can use this to obtain a similar result for conformal deformations near ∂FM. Corollary 3 Let ψF ∈ C∞(M)be a smooth function supported near ∂FMsuch that
ψF|∂FM is locally constant and consider the new Fc ∂-metricsg= eψFg, h= eψFh.
Then we have that
det(g, h)= det(g, h).
Proof We could use the Polyakov formula of [1], but this would require some extra decay behavior in ψF. Instead, we simply apply the previous theorem twice by un-doing and un-doing again a conformal surgery at each boundary component of ∂FMto
7 Compactness of Families of Relatively Isospectral Surfaces
As mentioned in the Introduction, Borthwick and Perry [6] prove a compactness the-orem for isoresonant metrics that coincide cocompactly and whose ends are hyper-bolic funnels. Their proof of compactness, like the proof of compactness of Osgood– Phillips–Sarnak, uses the spectral assumption only through the equality of the relative heat invariants (5.6) and relative determinants (6.1). We restate their theorem with these assumptions.
Theorem 5 (Borthwick–Perry [6]) Let (Mi, gi)be a family of Riemannian surfaces
that coincide cocompactly, whose ends are hyperbolic funnels, and assume that the relative heat invariants and relative determinants satisfy
ak(gi, gj)= 0, det(gi, gj)= 1, for all i, j, k.
Then there is a Riemannian surface (M, g∞), a subsequence (Mik, gik), and a
se-quence of diffeomorphisms
φk: M −→ Mik, with φ ◦ φ−1 |U∞= Id for any , such that the metrics φk∗gik converge to g∞inC∞.
We can now finally give a proof of our main result.
Proof of Theorem1 We need to first check that the various surfaces have the same topology. This can be deduced from the relative heat invariants. Namely, given two of the relatively isospectral surfaces (Mi, gi)and (Mk, gk), we can deform conformally
gi and gk inU∞in the same way to obtain metricsgi andgk also having vanishing relative heat invariants and such that
(i) the metricsgi andgk have no cusp, the cusps being removed by a conformal surgery at each marked point;
(ii) the metricsgi,gkdefine incomplete metrics on Miand Mkwith boundary having no geodesic curvature.
Since these metricsgi,gk have the same heat invariants, we know from [25] that Mi and Mkhave the same Euler characteristic, so the same topology. Thus, the isometry
Mi\ Ki→ Mk\ Kkcan be extended to a diffeomorphism Mi→ Mk. This means we can assume all the relatively isospectral metrics are defined on the same surface M∞ and agree onU∞= M \ K where K ⊂ M \ ∂M is a compact set.
Going back to the initial metrics gi, we can, by doing a conformal surgery onU∞, remove all the cusps and transform each boundary into a funnel end hyperbolic near infinity. We can also conformally modify the metrics in the remaining funnels to make them hyperbolic near infinity. We thus get a new sequence of metricsgi =
and a sequence of diffeomorphisms φik: M → Mik such that φi∗kgik converges tog∞
inC∞(M,g∞). As explained in [6], we can choose the diffeomorphisms such that
φik◦ φi−1k = Id near each funnel of (M,g∞). The same argument can be carried out near each (filled) cusp, so that the diffeomorphisms can be chosen so that φik◦ φi−1
k =
Id on U∞. Undoing the conformal transformation on the metric g∞ to obtain the metric g∞= e−ψg∞on M, we obtain the desired result with the subsequence gik
and the diffeomorphisms φik.
Acknowledgements The authors are grateful to David Borthwick, Gilles Carron, Andrew Hassell, Rafe Mazzeo, and Richard Melrose for helpful conversations and to the anonymous referee for their comments.
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